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• 2007 AIME II Problems/Problem 12
  The increasing [[geometric sequence]] <math>x_{0},x_{1},x_{2},\ldots</math> consists entirely of [[integer|inte ...<math>a^8r^{28} = 3^{308}</math>. Since all of the terms of the geometric sequence are integral powers of <math>3</math>, we know that both <math>a</math> and
  5 KB (829 words) - 12:22, 8 January 2024
• 2007 AMC 12A Problems
  ...c, d</math>, and <math>e</math> be five consecutive terms in an arithmetic sequence, and suppose that <math>a+b+c+d+e=30</math>. Which of <math>a, b, c, d,</ma ...d with the term 824. Let <math>S</math> be the sum of all the terms in the sequence. What is the largest [[prime]] [[factor]] that always divides <math>S</math
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 AMC 12A Problems/Problem 7
  ...c, d</math>, and <math>e</math> be five consecutive terms in an arithmetic sequence, and suppose that <math>a+b+c+d+e=30</math>. Which of <math>a, b, c, d,</ma
  818 bytes (152 words) - 16:40, 5 April 2024
• 2007 AMC 12A Problems/Problem 16
  We can find the number of increasing [[arithmetic sequence]]s of length 3 possible from 0 to 9, and then find all the possible permuta
  2 KB (336 words) - 05:01, 4 November 2022
• 2005 AMC 12A Problems/Problem 11
  Thus, the three digits form an [[arithmetic sequence]].
  2 KB (266 words) - 00:59, 19 October 2020
• 2005 AMC 12A Problems/Problem 13
  arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?
  3 KB (430 words) - 18:52, 11 July 2020
• 2006 AIME II Problems/Problem 13
  ...h>2n+1 + 2(j-1) = 2(n+j) - 1</math>. The odd integers form an [[arithmetic sequence]] with sum <math>N = j\left(\frac{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)< ...he <math>q</math>th positive odd number, and the largest odd number in the sequence be the <math>p</math>th positive odd number. Therefore, the sum is <math>p^
  4 KB (675 words) - 10:40, 14 July 2022
• 1999 AHSME Problems
  Define a sequence of real numbers <math>a_1, a_2, a_3, \ldots</math> by <math>a_1 = 1</math> The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99
  13 KB (1,945 words) - 18:28, 19 June 2023
• Aleph null
  ...erms in any [[arithmetic sequence]], [[geometric sequence]], or [[harmonic sequence]]. It is less than, for example, aleph 1 (<math>\aleph_{1}</math>), which i
  847 bytes (120 words) - 20:49, 26 October 2007
• 2004 AMC 12A Problems/Problem 14
  A [[sequence]] of three real numbers forms an [[arithmetic progression]] with a first te ...is <math>9</math>, <math>9+d</math>, and <math>9+2d</math>. The geometric sequence (when expressed in terms of <math>d</math>) has the terms <math>9</math>, <
  4 KB (689 words) - 03:35, 16 January 2023
• 1990 USAMO Problems/Problem 4
  Let a <math>k</math>-good sequence be a sequence of distinct integers <math>\{ a_i \}_{i=1}^k</math> such that for all integ ...ood sequence, then <math>\{ a_i \}_{i=1}^k</math> is a <math>k</math>-good sequence which starts on <math>a</math>, so it is a permutation of <math>k</math> co
  3 KB (529 words) - 19:15, 18 July 2016
• 1959 AHSME Problems
  A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let S be the sum of the first nine terms of the sequence <math>x+a, x^2+2a, x^3+3a, \cdots.</math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  In a given arithmetic sequence the first term is <math>2</math>, the last term is <math>29</math>, and the ...>s_2</math> be the sum of the first <math>n</math> terms of the arithmetic sequence <math>17,19,\cdots</math>. Assume <math>n \ne 0</math>. Then <math>s_1=s_2<
  19 KB (3,159 words) - 22:10, 11 March 2024
• 2002 AMC 12B Problems/Problem 5
  ...ath> and <math>v, w, x, y, </math> and <math>z</math> form an [[arithmetic sequence]]. Find the value of <math>x</math>. ...etic sequence with an odd number of terms, it is simply the average of the sequence.
  2 KB (263 words) - 05:58, 8 August 2023
• 2002 AMC 12B Problems/Problem 9
  ...an increasing arithmetic sequence and <math>a,b,d</math> form a geometric sequence, then <math>\frac ad</math> is As <math>a, b, d</math> is a geometric sequence, let <math>b=ka</math> and <math>d=k^2a</math> for some <math>k>0</math>.
  2 KB (288 words) - 21:42, 11 December 2017
• 2002 AMC 12B Problems/Problem 10
  The set is an arithmetic sequence of numbers each <math>1</math> more than a multiple of <math>3</math>. Thus
  1 KB (166 words) - 00:43, 17 January 2021
• 1995 USAMO Problems
  ...n \,</math> is the least positive integer that does not form an arithmetic sequence of length <math>\, p \,</math> with any of the preceding terms. Prove that, ...itive rational number <math>\, q, \,</math> show that pressing some finite sequence of buttons will yield <math>\, q</math>. Assume that the calculator does
  3 KB (540 words) - 13:31, 4 July 2013
• Partition (combinatorics)
  ...bers is through [[generating function]]s. The generating function for the sequence <math>\{P(n)\}_{n \geq 0}</math> is given by <math>F(x)= \sum_{n \geq 0}P(n Using the formula for the sum of an [[infinite]] [[geometric sequence]] we can express this in the more compact form
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2008 AMC 12A Problems
  ...s of an arithmetic sequence, and the <math>12^\text{th}</math> term of the sequence is <math>\log(b^n)</math>. What is <math>n</math>? Let <math>a_1,a_2,\ldots</math> be a sequence determined by the rule <math>a_n=a_{n-1}/2</math> if <math>a_{n-1}</math> i
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 16
  ...an [[arithmetic sequence]], and the <math>12^\text{th}</math> term of the sequence is <math>\log{b^n}</math>. What is <math>n</math>? The first three terms of the arithmetic sequence are <math>3A + 7B</math>, <math>5A + 12B</math>, and <math>8A + 15B</math>,
  3 KB (577 words) - 16:33, 9 October 2022
• 2008 AMC 12B Problems/Problem 24
  ...tween the x coordinates of consecutive <math>A_i</math> form an arithmetic sequence (<math>x_{A_1} - x_{A_0} = \frac{2}{3}</math>, <math>x_{A_2} - x_{A_1} = \f
  9 KB (1,482 words) - 13:52, 4 April 2024
• Symmetric group
  ...esentations of permutations: as functions, as products of [[cycle]]s, as [[sequence]]s or [[word]]s, etc.) Knowledge of the symmetric group <math>S_{n}</math>
  10 KB (1,668 words) - 15:33, 25 May 2008
• 2008 AIME II Problems/Problem 1
  Rewriting this sequence with more terms, we have ...(a+1)^2 - a^2 = 8a + 12</cmath>, turning <math>N</math> into a arithmetic sequence with 25 terms, them being <math>1, 5, 9, \dots ,97</math>, as the series <m
  4 KB (575 words) - 16:41, 14 April 2024
• 2008 Mock ARML 1 Problems
  ...g it with one of the opposite color. Compute the probability that, after a sequence of turns, there are <math>5</math> black balls in the hat before there are
  3 KB (409 words) - 16:41, 29 May 2008
• 2008 iTest Problems
  ...st <math>1</math> that he rolled. His first <math>31</math> rolls make the sequence <math>4,3,11,3,11,8,5,2,12,9,5,7,11,3,6,10,\textbf{1},8,3,\textbf{2},10,4,2 ...ng - do <math>\textit{not}</math> assume he starts by rolling the specific sequence of <math>31</math> rolls above.)
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 94
  sequence ...7</math> and <math>2008</math> must be a divisor of some term in the given sequence. The largest prime less than <math>2008</math> is <math>2003</math>, which
  4 KB (571 words) - 21:21, 22 November 2018
• Prime triplet
  A set of three [[prime number]]s which form an arithmetic sequence with common difference two is called a '''prime triplet'''.
  836 bytes (121 words) - 00:59, 17 March 2009
• 1973 USAMO Problems/Problem 5
  ...t the cube roots of three distinct prime numbers cannot form an arithmetic sequence.
  4 KB (683 words) - 20:18, 29 December 2019
• 2002 AMC 10B Problems
  Suppose that <math>\{a_n\}</math> is an arithmetic sequence with Let <math>\{a_k\}</math> be a sequence of integers such that <math>a_1=1</math> and <math>a_{m+n}=a_m+a_n+mn,</mat
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2004 AMC 10B Problems
  In the sequence <math>2001</math>, <math>2002</math>, <math>2003</math>, <math>\ldots</math <math>2004^\textrm{th}</math> term in this sequence?
  13 KB (1,988 words) - 20:19, 15 May 2024
• 2000 AMC 10 Problems
  ...is the last to appear in the units position of a number in the Fibonacci sequence?
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1999 AHSME Problems/Problem 20
  The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99
  2 KB (268 words) - 14:00, 21 March 2023
• 1985 AJHSME Problems/Problem 2
  ...2}\cdot(a_1+a_n)</math> where <math>n</math> is the number of terms in the sequence, <math>a_1</math> is the first term and <math>a_n</math> is the last term.
  2 KB (282 words) - 13:43, 4 April 2024
• 1986 AJHSME Problems/Problem 25
  ...the largest (since the sum of the 2 entries is twice the average of whole sequence). <math>2+100=102</math>, <math>3+99=102</math>, <math>4+100=104</math>, <m
  2 KB (291 words) - 20:13, 17 January 2024
• 2004 AMC 10B Problems/Problem 21
  ...t <math>S</math> is the union of the first <math>2004</math> terms of each sequence. How many distinct numbers are in <math>S</math>?
  2 KB (357 words) - 16:20, 5 May 2024
• 2009 AMC 12A Problems
  ...h>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>? ..._2</math>, <math>F_3</math>, and <math>F_4</math> shown are the first in a sequence of figures. For <math>n\ge3</math>, <math>F_n</math> is constructed from <m
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2009 AMC 12A Problems/Problem 7
  ...h>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>? As this is an arithmetic sequence, the difference must be constant: <math>(5x-11) - (2x-3) = (3x+1) - (5x-11)
  825 bytes (128 words) - 10:17, 9 February 2015
• 1988 AJHSME Problems
  What is the <math>100\text{th}</math> number in the arithmetic sequence: <math>1,5,9,13,17,21,25,...</math>?
  14 KB (1,872 words) - 15:23, 17 January 2023
• 2009 AIME I Problems/Problem 7
  The sequence <math>(a_n)</math> satisfies <math>a_1 = 1</math> and <math>5^{(a_{n + 1} - Plug in <math>n = 1, 2, 3, 4</math> to see the first few terms of the sequence: <cmath>\log_5{5},\log_5{8}, \log_5{11}, \log_5{14}.</cmath> We notice that
  2 KB (340 words) - 00:26, 9 January 2023
• 1988 AJHSME Problems/Problem 19
  What is the <math>100\text{th}</math> number in the [[arithmetic sequence]]: <math>1,5,9,13,17,21,25,...</math>? To get from the <math>1^\text{st}</math> term of an arithmetic sequence to the <math>100^\text{th}</math> term, we must add the common [[difference
  946 bytes (133 words) - 10:51, 28 June 2023
• 2009 IMO Problems/Problem 3
  Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequences are both arithmetic progressions. Prove that the sequence <math>s_1,s_2,s_3,\ldots</math> is itself an arithmetic progression.
  1 KB (184 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequences are both arithmetic progressions. Prove that the sequence <math>s_1,s_2,s_3,\ldots</math> is itself an arithmetic progression.
  3 KB (509 words) - 09:23, 10 September 2020
• 1995 USAMO Problems/Problem 1
  ...n \,</math> is the least positive integer that does not form an arithmetic sequence of length <math>\, p \,</math> with any of the preceding terms. Prove that, ...n that <math>a_{n+1}>a_{n}</math> (without this assumption, I can have the sequence
  4 KB (625 words) - 18:23, 22 March 2024
• Unique factorization domain
  So now we can construct a sequence <math>r_1,r_2,\ldots</math> of elements of <math>S</math> such that
  6 KB (1,217 words) - 23:05, 23 August 2009
• 2010 AMC 12A Problems
  ...nd <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence?
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 12A Problems/Problem 20
  ...integer terms with <math>a_1=b_1=1</math>, we can write the terms of each sequence as ...th>(m-1)</math> times the common difference for that particular arithmetic sequence. Let the common difference of <math>(a_n)</math> be <math>k</math> and the
  5 KB (797 words) - 15:27, 3 July 2023
• 2010 AMC 12A Problems/Problem 10
  ...nd <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence?
  1 KB (178 words) - 20:47, 27 October 2022
• 2010 AMC 12A Problems/Problem 23
  ...liminated. After factoring out a 2 from each of the 9 even numbers in this sequence, the 10, 20, 30, ..., 90 becomes 1, 2, 3, 4, 1, 6, 7, 8, 9, whose product i
  10 KB (1,525 words) - 09:44, 24 April 2024
• 2010 AMC 10B Problems
  ...the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neithe
  12 KB (1,817 words) - 22:44, 22 December 2020
• 2010 AMC 12B Problems
  ...the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neithe A geometric sequence <math>(a_n)</math> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, an
  12 KB (1,845 words) - 13:00, 19 February 2020

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